72 III METRIC SPACES

It is immediately verified that this function satisfies the axioms (I) to (IV)
in Section 3.1 , in other words, it is a distance on E; the metric space obtained
by taking d as a distance on E is called the product of the two metric spaces
E1? E2 . The mapping (xi9 x2) -»• (x2 , *i) of E1 x E2 onto E2 x EA is an
isometry.

We observe that the two functions d', d" defined by

d'(x, y) = d&i 9yJ + ^2(x2 , J>2)

are also distances on E, as is easily verified, and are uniformly equivalent
to d (see Section 3.14), for we have

d(x, y) < d*(x, y) < d'(x9 y} < 2rf(*, j;).

For all questions dealing with topological properties (or Cauchy sequences
and uniformly continuous functions) it is therefore equivalent to take on E
any one of the distances d, d'9 d". When nothing is said to the contrary, we
will consider on E the distance d. Open (resp. closed) balls for the distances
d, dl9 d2 will be respectively written B, Bl9 B2 (resp. B', B'l9 B2) instead of the
uniform notation B (resp. B') used up to now.

(3.20.1) For any point a = (aly a2) e E and any r > 0, we have B(#; r) =
BI(*I; r) x B2fe; r) and W(a; r) = E[(a,; r) x Bi(a2; r).

This follows at once from the definition of d.

(3.20.2) If AI is an open set in El9 A2 an open set in E2, then Al x A2
is open in Ex x E2.

For if a = (al9 a2) e Ax x A2, there exists rl > 0 and r2 > 0 such
that B!^; rx) c A1? B2(^2; r2) c A2; take r = mm(rl5 r2); then by (3.20.1),
B(a; r) c: A! x A2.

(3.20.3) For any pair of sets Aj c El9 A2 e E2, Ax x A2 = AA x A2; m
particular, in order that AJ x A2 be closed in E, a necessary and sufficient
condition is that Ax be closed in Ei and A2 closed in E2.

If a = (al9 a2) e A1 x A2, for any e > 0 there is, by assumption, an jq e A1
and an x2 e A2 such that dfot, xj < 8, d2(a2, x2) < e; hence if x = (jq, x2),
pact subspace of E, which is not locally connected; the connected components of E are